Seymour\u27s second-neighborhood conjecture from a different perspective

Seymour\u27s Second-Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex (Formula presented.) such that the number of vertices of out-distance two from (Formula presented.) is at least as large as the number of vertices of out-distance one from it. We present alternative statements of the conjecture in the language of linear algebra

07281 Abstracts Collection -- Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs

From 8th to 13th July 2007, the Dagstuhl Seminar ``Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Reliability of Partial k-tree Networks

133 pagesRecent developments in graph theory have shown the importance of the class of partial k- trees. This large class of graphs admits several algorithm design methodologies that render efficient solutions for a large number of problems inherently difficult for general graphs. In this thesis we develop such algorithms to solve a variety of reliability problems on partial k-tree networks with node and edge failures. We also investigate the problem of designing uniformly optimal 2-trees with respect to the 2-terminal reliability measure. We model a. communication network as a graph in which nodes represent communication sites and edges represent bidirectional communication lines. Each component (node or edge) has an associated probability of operation. Components of the network are in either operational or failed state and their failures are statistically independent. Under this model, the reliability of a network G is defined as the probability that a given connectivity condition holds. The l-terminal reliability of G, Rel1 ( G), is the probability that any two of a given set of I nodes of G can communicate. Robustness of a network to withstand failures can be expressed through network resilience, Res( G), which is the expected number of distinct pairs of nodes that can communicate. Computing these and other similarly defined measures is #P-hard for general networks. We use a dynamic programming paradigm to design linear time algorithms that compute Rel1( G), Res( G), and some other reliability and resilience measures of a partial k-tree network given with an embedding in a k-tree (for a fixed k). Reliability problems on directed networks are also inherently difficult. We present efficient algorithms for directed versions of typical reliability and resilience problems restricted to partial k-tree networks without node failures. Then we reduce to those reliability problems allowing both node and edge failures. Finally, we study 2-terminal reliability aspects of 2-trees. We characterize uniformly optimal 2-trees, 2-paths, and 2-caterpillars with respect to Rel2 and identify local graph operations that improve the 2-terminal reliability of 2-tree networks

Well-Quasi-Ordering by the Induced-Minor Relation

Robertson and Seymour proved Wagner\u27s Conjecture, which says that finite graphs are well-quasi-ordered by the minor relation. Their work motivates the question as to whether any class of graphs is well-quasi-ordered by other containment relations. This dissertation is concerned with a special graph containment relation, the induced-minor relation. This dissertation begins with a brief introduction to various graph containment relations and their connections with well-quasi-ordering. In the first chapter, we discuss the results about well-quasi-ordering by graph containment relations and the main problems of this dissertation. The graph theory terminology and preliminary results that will be used are presented in the next chapter. The class of graphs that is considered in this research is the class W of graphs that contain neither W4 (a wheel graph with five vertices) and K5\e (a complete graph on five vertices minus an edge) as an induced minor. Chapter 3 is devoted to studying the structure of this class of graphs. A class of graphs is well-quasi-ordered by a containment relation if it contains no infinite antichain, so infinite antichains are important. We construct in Chapter 4 an infinite antichain of W with respect to the induced minor relation and study its important properties in Chapter 5. These properties are used in determining all well-quasi-ordered subclasses of W to reach the main result of Chapter 6

Group Connectivity of Graphs

Tutte introduced the theory of nowhere-zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere-zero A-flow, for any Abelian group A with |A| ≥ k. In 1992 Jaeger et al. [16] extended nowhere-zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b: V (G) A with sumv∈ V(G ) b(v) = 0, there always exists a map ƒ: E(G) A - {lcub}0{rcub}, such that at each v ∈ V(G), e=vw isdirectedfrom vtow fe- e=uvi sdirectedfrom utov fe=b v in A, then G is A-connected. For a 2-edge-connected graph G, define Lambda g(G) = min{lcub}k: for any Abelian group A with |A| ≥ k, G is A-connected{rcub}.;Let G1 ⊗ G2 and G1 xG2 denote the strong and Cartesian product of two connected nontrivial graphs G1 and G2. We prove that Lambdag(G 1 ⊗ G2) ≤ 4, where equality holds if and only if both G1 and G 2 are trees and min{lcub}|V (G1)|, |V (G2)|{rcub}=2; Lambda g(G1 ⊗ G 2) ≤ 5, where equality holds if and only if both G 1 and G2 are trees and either G 1 ≅ K1, m and G2 ≅ K 1,n, for n, m ≥ 2 or min{lcub}|V (G1)|, | V (G2)|{rcub}=2. A similar result for the lexicographical product graphs is also obtained.;Let P denote a path in G, let beta G(P) be the minimum length of a circuit containing P, and let betai(G) be the maximum of betaG(P) over paths of length i in G. We show that Lambda g(G) ≤ betai( G) + 1 for any integer i \u3e 0 and for any 2-connected graph G. Partial solutions toward determining the graphs for which equality holds were obtained by Fan et al. in [J. Comb. Theory, Ser. B, 98(6) (2008), 1325-1336], among others. We completely determine all graphs G with Lambda g(G) = beta2(G) + 1.;Let Z3 denote the cyclic group of order 3. In [16], Jaeger et al. conjectured that every 5-edge-connected graph is Z3 -connected. We proved the following: (i) Every 5-edge-connected graph is Z3 -connected if and only if every 5-edge-connected line graph is Z3 -connected. (ii) Every 6-edge-connected triangular line graph is Z3 -connected. (iii) Every 7-edge-connected triangular claw-free graph is Z3 -connected. In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere-zero 3-flow

The bidimensionality theory and its algorithmic applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 201-219).Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(k²), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results.(cont.) The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L₁ (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O[sq. root( log n)] approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be [theta][sq. root(log n)]. We also prove various approximate max-flow/min-vertex- cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O[sq. root (log n)] pseudo-approximation for finding balanced vertex separators in general graphs.(cont.) Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k[sq. root(log k)]), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.by MohammadTaghi Hajiaghayi.Ph.D